Matrices de rotaciones, simetr\'{\i}as y roto-simetr\'{\i}as

Abstract

In this note we find the orthogonal matrices $R,S\in M_3(\mathbb{R})$ corresponding to the clockwise rotation $r$ in $\mathbb{R}^3$ around the axis generated by a unit vector $u=(a,b,c)^t$ through an angle $\alpha\in [0,2\pi)$, and to the symmetry $s$ in $\mathbb{R}^3$ on the plane perpendicular to $u$. Matrix $S$ depends on $a,b,c$ and matrix $R$ depends on $a,b,c, \cos \alpha$ and $\sin \alpha$. We show $SR=RS$. The matrix $R$ is due to Alperin.